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Properties

Label 2-2352-84.47-c0-0-2
Degree $2$
Conductor $2352$
Sign $-0.0633 - 0.997i$
Analytic cond. $1.17380$
Root an. cond. $1.08342$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)9-s + 1.73i·13-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)25-s − 0.999·27-s + (−0.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s + (−1.49 + 0.866i)39-s + 1.73i·43-s + 0.999·57-s + (1.5 − 0.866i)67-s + (1.5 − 0.866i)73-s + (−0.499 + 0.866i)75-s + (−1.5 − 0.866i)79-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)9-s + 1.73i·13-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)25-s − 0.999·27-s + (−0.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s + (−1.49 + 0.866i)39-s + 1.73i·43-s + 0.999·57-s + (1.5 − 0.866i)67-s + (1.5 − 0.866i)73-s + (−0.499 + 0.866i)75-s + (−1.5 − 0.866i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.0633 - 0.997i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (1391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :0),\ -0.0633 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.329148848\)
\(L(\frac12)\) \(\approx\) \(1.329148848\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - 1.73iT - T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - 1.73iT - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.386725719406232762259228325723, −8.829431542911691174532891929046, −7.943640985036925301273516970132, −7.10305047058641789067305016253, −6.30792394344717580039821369900, −5.16571731499131676393689109910, −4.57264677772177362081331077248, −3.72928945178457173015163671959, −2.81490086691775600320431139664, −1.74336759877131082053515560156, 0.876804002733165351353712806347, 2.15346985224634997342782038706, 3.11110430347519644808400840119, 3.82989702312932093486584324411, 5.29843358412518647895401140583, 5.79350615601559685751274934500, 6.85300142354819138364791964514, 7.44131565601255858007011060816, 8.285130916565853550874192986572, 8.632883593553366184180235509258

Graph of the $Z$-function along the critical line